2021
DOI: 10.1007/s10468-021-10071-9
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Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories

Abstract: We prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander–Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.

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“…In [23] and [10], Butler and Auslander showed that the relations in the Grothendieck group of an Artin algebra are generated by the ones given by almost‐split sequences precisely when the algebra is of finite representation type. This result was generalized to triangulated categories with certain finiteness conditions by Xiao and Zhu in [101], to Hom$\operatorname{Hom}$‐finite, Krull–Schmidt exact categories with enough projectives by Enomoto in [33], to false(n+2false)$(n+2)$‐angulated categories by Fedele in [34], and to Hom$\operatorname{Hom}$‐finite, Krull–Schmidt triangulated categories with a cogenerator by Haugland in [45]. Relations in the Grothendieck group of cluster categories have also been studied in [76].…”
Section: Relations For G${{g}}$‐vectors In Finite‐type Cluster Algebr...mentioning
confidence: 99%
“…In [23] and [10], Butler and Auslander showed that the relations in the Grothendieck group of an Artin algebra are generated by the ones given by almost‐split sequences precisely when the algebra is of finite representation type. This result was generalized to triangulated categories with certain finiteness conditions by Xiao and Zhu in [101], to Hom$\operatorname{Hom}$‐finite, Krull–Schmidt exact categories with enough projectives by Enomoto in [33], to false(n+2false)$(n+2)$‐angulated categories by Fedele in [34], and to Hom$\operatorname{Hom}$‐finite, Krull–Schmidt triangulated categories with a cogenerator by Haugland in [45]. Relations in the Grothendieck group of cluster categories have also been studied in [76].…”
Section: Relations For G${{g}}$‐vectors In Finite‐type Cluster Algebr...mentioning
confidence: 99%